Analytically solving a 1st order quadratic nonlinear ODE system I sincerely wish somebody can help me to analytically solve following ODE system. $x(t)$ $y(t)$ $z(t)$ are 3 functions of $t$, and the system is derived from an improved version of multi-compartment model:
$$\left\{ \begin{array}{l}
x^\prime(t)=A \cdot x(t)^2+B \cdot x+C \cdot y+D \cdot z\\
y^\prime(t)=E \cdot y(t)^2+F \cdot x+G \cdot y+H \cdot z\\
z^\prime(t)=I \cdot z(t)^2+G \cdot x+K \cdot y+L \cdot z
\end{array} \right.$$
A,B,C..., are just any constants, so its asymmetric.
And furthermore, when the constants also become functions of t, above system becomes
$$\left\{ \begin{array}{l}
x^\prime(t)=A(t) \cdot x(t)^2+B(t) \cdot x(t)+C(t) \cdot y(t)+D(t) \cdot z(t)\\
y^\prime(t)=E(t) \cdot y(t)^2+F(t) \cdot x(t)+G(t) \cdot y(t)+H(t) \cdot z(t)\\
z^\prime(t)=I(t) \cdot z(t)^2+G(t) \cdot x(t)+K(t) \cdot y(t)+L(t) \cdot z(t)
\end{array} \right.$$
I need the close-forms of $x(t)$ $y(t)$ $z(t)$ of both two ODE systems, so that I can do my later tasks... I will appreciate you a lot if you can offer me any solutions, hopefully a general solution for any $n$.
I was hoping to solve them by learning Riccati quadratic ODEs, but due to my limited knowledge I couldn't find solutions yet. So if you can point me any good materials, I will also appreciate. : )
Thanks a lot!!!!
 A: As I have pointed out in my comment, this system can be reduced to a more treatable one. Once you have cast it into the form
$$\left\{ \begin{array}{l}
\xi^\prime=\xi^2+b\xi+c\eta+d\theta\\
\eta^\prime=\eta^2+f\xi+g\eta+h\theta\\
\theta^\prime=\theta^2+j\xi+k\eta+l\theta
\end{array} \right.$$
one introduces the three new variables
$$\left\{ \begin{array}{l}
   \xi=-\frac{1}{X}\frac{dX}{dt} \\
   \eta=-\frac{1}{Y}\frac{dY}{dt} \\
   \theta=-\frac{1}{Z}\frac{dZ}{dt}
\end{array} \right.$$
and you will remove the quadratic terms, obtaining
$$\left\{ \begin{array}{l}
   \frac{1}{X}\frac{d^2X}{dt^2}=b\frac{1}{X}\frac{dX}{dt}+c\frac{1}{Y}\frac{dY}{dt}+d\frac{1}{Z}\frac{dZ}{dt} \\
   \frac{1}{Y}\frac{d^2Y}{dt^2}=f\frac{1}{X}\frac{dX}{dt}+g\frac{1}{Y}\frac{dY}{dt}+h\frac{1}{Z}\frac{dZ}{dt} \\
   \frac{1}{Z}\frac{d^2Z}{dt^2}=j\frac{1}{X}\frac{dX}{dt}+k\frac{1}{Y}\frac{dY}{dt}+l\frac{1}{Z}\frac{dZ}{dt}
\end{array} \right.$$
This system can be easily rewritten as
$$\left\{ \begin{array}{l}
   \frac{d^2X}{dt^2}=b\frac{dX}{dt}+X\frac{d}{dt}\ln(Y^cZ^d) \\
   \frac{d^2Y}{dt^2}=g\frac{dY}{dt}+Y\frac{d}{dt}\ln(X^fZ^h) \\
   \frac{d^2Z}{dt^2}=l\frac{dZ}{dt}+Z\frac{d}{dt}\ln(X^jY^k)
\end{array} \right.$$
that can be solved with standard exponentials provided a relation between the coefficients exists so that the derivatives of the logarithms are zero.
For the more general case I cannot be of help but one can always reduce to a similar set of equations.
